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Theorem List for Metamath Proof Explorer - 37901-38000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxphe 37901 Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) (Proof shortened by OpenAI, 3-Jul-2020.)
(𝐴 × 𝐵) hereditary 𝐵

Theorem0he 37902 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
∅ hereditary 𝐴

Theorem0heALT 37903 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
∅ hereditary 𝐴

Theoremhe0 37904 Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
𝐴 hereditary ∅

Theoremunhe1 37905 The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.)
((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → (𝑅𝑆) hereditary 𝐴)

Theoremsnhesn 37906 Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
{⟨𝐴, 𝐴⟩} hereditary {𝐵}

Theoremidhe 37907 The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
I hereditary 𝐴

Theorempsshepw 37908 The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
[] hereditary 𝒫 𝐴

Theoremsshepw 37909 The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
( [] ∪ I ) hereditary 𝒫 𝐴

20.27.3.3  _Begriffsschrift_ Chapter II Implication

Axiomax-frege1 37910 The case in which 𝜑 is denied, 𝜓 is affirmed, and 𝜑 is affirmed is excluded. This is evident since 𝜑 cannot at the same time be denied and affirmed. Axiom 1 of [Frege1879] p. 26. Identical to ax-1 6. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Axiomax-frege2 37911 If a proposition 𝜒 is a necessary consequence of two propositions 𝜓 and 𝜑 and one of those, 𝜓, is in turn a necessary consequence of the other, 𝜑, then the proposition 𝜒 is a necessary consequence of the latter one, 𝜑, alone. Axiom 2 of [Frege1879] p. 26. Identical to ax-2 7. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Theoremrp-simp2-frege 37912 Simplification of triple conjunction. Compare with simp2 1061. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜓)))

Theoremrp-simp2 37913 Simplification of triple conjunction. Identical to simp2 1061. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓𝜒) → 𝜓)

Theoremrp-frege3g 37914 Add antecedent to ax-frege2 37911. More general statement than frege3 37915. Like ax-frege2 37911, it is essentially a closed form of mpd 15, however it has an extra antecedent.

It would be more natural to prove from a1i 11 and ax-frege2 37911 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

(𝜑 → ((𝜓 → (𝜒𝜃)) → ((𝜓𝜒) → (𝜓𝜃))))

Theoremfrege3 37915 Add antecedent to ax-frege2 37911. Special case of rp-frege3g 37914. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒 → (𝜑𝜓)) → ((𝜒𝜑) → (𝜒𝜓))))

Theoremrp-misc1-frege 37916 Double-use of ax-frege2 37911. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑 → (𝜓𝜒)) → (𝜑𝜓)) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))

Theoremrp-frege24 37917 Introducing an embedded antecedent. Alternate proof for frege24 37935. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))

Theoremrp-frege4g 37918 Deduction related to distribution. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜑 → ((𝜓𝜒) → (𝜓𝜃))))

Theoremfrege4 37919 Special case of closed form of a2d 29. Special case of rp-frege4g 37918. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (𝜒 → (𝜑𝜓))) → ((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓))))

Theoremfrege5 37920 A closed form of syl 17. Identical to imim2 58. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theoremrp-7frege 37921 Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓𝜒)) → (𝜃 → ((𝜑𝜓) → (𝜑𝜒))))

Theoremrp-4frege 37922 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
((𝜑 → ((𝜓𝜑) → 𝜒)) → (𝜑𝜒))

Theoremrp-6frege 37923 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
(𝜑 → ((𝜓 → ((𝜒𝜓) → 𝜃)) → (𝜓𝜃)))

Theoremrp-8frege 37924 Eliminate antecedent when it is implied by previous antecedent. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓 → ((𝜒𝜓) → 𝜃))) → (𝜑 → (𝜓𝜃)))

Theoremrp-frege25 37925 Closed form for a1dd 50. Alternate route to Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))

Theoremfrege6 37926 A closed form of imim2d 57 which is a deduction adding nested antecedents. Proposition 6 of [Frege1879] p. 33. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → ((𝜃𝜓) → (𝜃𝜒))))

Theoremaxfrege8 37927 Swap antecedents. Identical to pm2.04 90. This demonstrates that Axiom 8 of [Frege1879] p. 35 is redundant.

Proof follows closely proof of pm2.04 90 in http://us.metamath.org/mmsolitaire/pmproofs.txt, but in the style of Frege's 1879 work. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Theoremfrege7 37928 A closed form of syl6 35. The first antecedent is used to replace the consequent of the second antecedent. Proposition 7 of [Frege1879] p. 34. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒 → (𝜃𝜑)) → (𝜒 → (𝜃𝜓))))

Axiomax-frege8 37929 Swap antecedents. If two conditions have a proposition as a consequence, their order is immaterial. Third axiom of Frege's 1879 work but identical to pm2.04 90 which can be proved from only ax-mp 5, ax-frege1 37910, and ax-frege2 37911. (Redundant) Axiom 8 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Theoremfrege26 37930 Identical to idd 24. Proposition 26 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜓))

Theoremfrege27 37931 We cannot (at the same time) affirm 𝜑 and deny 𝜑. Identical to id 22. Proposition 27 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑𝜑)

Theoremfrege9 37932 Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 37920 only in an unessential way. Identical to imim1 83. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremfrege12 37933 A closed form of com23 86. Proposition 12 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜑 → (𝜒 → (𝜓𝜃))))

Theoremfrege11 37934 Elimination of a nested antecedent as a partial converse of ja 173. If the proposition that 𝜓 takes place or 𝜑 does not is a sufficient condition for 𝜒, then 𝜓 by itself is a sufficient condition for 𝜒. Identical to jarr 106. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))

Theoremfrege24 37935 Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 37917 which was proved without relying on ax-frege8 37929. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))

Theoremfrege16 37936 A closed form of com34 91. Proposition 16 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃𝜏)))) → (𝜑 → (𝜓 → (𝜃 → (𝜒𝜏)))))

Theoremfrege25 37937 Closed form for a1dd 50. Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))

Theoremfrege18 37938 Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜃𝜑) → (𝜓 → (𝜃𝜒))))

Theoremfrege22 37939 A closed form of com45 97. Proposition 22 of [Frege1879] p. 41. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))) → (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜃𝜂))))))

Theoremfrege10 37940 Result commuting antecedents within an antecedent. Proposition 10 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑 → (𝜓𝜒)) → 𝜃) → ((𝜓 → (𝜑𝜒)) → 𝜃))

Theoremfrege17 37941 A closed form of com3l 89. Proposition 17 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜓 → (𝜒 → (𝜑𝜃))))

Theoremfrege13 37942 A closed form of com3r 87. Proposition 13 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜒 → (𝜑 → (𝜓𝜃))))

Theoremfrege14 37943 Closed form of a deduction based on com3r 87. Proposition 14 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃𝜏)))) → (𝜑 → (𝜃 → (𝜓 → (𝜒𝜏)))))

Theoremfrege19 37944 A closed form of syl6 35. Proposition 19 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜒𝜃) → (𝜑 → (𝜓𝜃))))

Theoremfrege23 37945 Syllogism followed by rotation of three antecedents. Proposition 23 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → ((𝜏𝜑) → (𝜓 → (𝜒 → (𝜏𝜃)))))

Theoremfrege15 37946 A closed form of com4r 94. Proposition 15 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃𝜏)))) → (𝜃 → (𝜑 → (𝜓 → (𝜒𝜏)))))

Theoremfrege21 37947 Replace antecedent in antecedent. Proposition 21 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → 𝜒) → ((𝜑𝜃) → ((𝜃𝜓) → 𝜒)))

Theoremfrege20 37948 A closed form of syl8 76. Proposition 20 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → ((𝜃𝜏) → (𝜑 → (𝜓 → (𝜒𝜏)))))

20.27.3.4  _Begriffsschrift_ Chapter II Implication and Negation

Theoremaxfrege28 37949 Contraposition. Identical to con3 149. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Axiomax-frege28 37950 Contraposition. Identical to con3 149. Theorem *2.16 of [WhiteheadRussell] p. 103. Axiom 28 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Theoremfrege29 37951 Closed form of con3d 148. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))

Theoremfrege30 37952 Commuted, closed form of con3d 148. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))

Theoremaxfrege31 37953 Identical to notnotr 125. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.)
(¬ ¬ 𝜑𝜑)

Axiomax-frege31 37954 𝜑 cannot be denied and (at the same time ) ¬ ¬ 𝜑 affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnotr 125. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)

Theoremfrege32 37955 Deduce con1 143 from con3 149. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))

Theoremfrege33 37956 If 𝜑 or 𝜓 takes place, then 𝜓 or 𝜑 takes place. Identical to con1 143. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → (¬ 𝜓𝜑))

Theoremfrege34 37957 If as a conseqence of the occurence of the circumstance 𝜑, when the obstacle 𝜓 is removed, 𝜒 takes place, then from the circumstance that 𝜒 does not take place while 𝜑 occurs the occurence of the obstacle 𝜓 can be inferred. Closed form of con1d 139. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓)))

Theoremfrege35 37958 Commuted, closed form of con1d 139. Proposition 35 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → (¬ 𝜒 → (𝜑𝜓)))

Theoremfrege36 37959 The case in which 𝜓 is denied, ¬ 𝜑 is affirmed, and 𝜑 is affirmed does not occur. If 𝜑 occurs, then (at least) one of the two, 𝜑 or 𝜓, takes place (no matter what 𝜓 might be). Identical to pm2.24 121. Proposition 36 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (¬ 𝜑𝜓))

Theoremfrege37 37960 If 𝜒 is a necessary consequence of the occurrence of 𝜓 or 𝜑, then 𝜒 is a necessary consequence of 𝜑 alone. Similar to a closed form of orcs 409. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → 𝜒) → (𝜑𝜒))

Theoremfrege38 37961 Identical to pm2.21 120. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝜑 → (𝜑𝜓))

Theoremfrege39 37962 Syllogism between pm2.18 122 and pm2.24 121. Proposition 39 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜑) → (¬ 𝜑𝜓))

Theoremfrege40 37963 Anything implies pm2.18 122. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝜑 → ((¬ 𝜓𝜓) → 𝜓))

Theoremaxfrege41 37964 Identical to notnot 136. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.)
(𝜑 → ¬ ¬ 𝜑)

Axiomax-frege41 37965 The affirmation of 𝜑 denies the denial of 𝜑. Identical to notnot 136. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝜑 → ¬ ¬ 𝜑)

Theoremfrege42 37966 Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
¬ ¬ (𝜑𝜑)

Theoremfrege43 37967 If there is a choice only between 𝜑 and 𝜑, then 𝜑 takes place. Identical to pm2.18 122. Proposition 43 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜑) → 𝜑)

Theoremfrege44 37968 Similar to a commuted pm2.62 425. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜓𝜑) → 𝜑))

Theoremfrege45 37969 Deduce pm2.6 182 from con1 143. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → (¬ 𝜓𝜑)) → ((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓)))

Theoremfrege46 37970 If 𝜓 holds when 𝜑 occurs as well as when 𝜑 does not occur, then 𝜓 holds. If 𝜓 or 𝜑 occurs and if the occurences of 𝜑 has 𝜓 as a necessary consequence, then 𝜓 takes place. Identical to pm2.6 182. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓))

Theoremfrege47 37971 Deduce consequence follows from either path implied by a disjunction. If 𝜑, as well as 𝜓 is sufficient condition for 𝜒 and 𝜓 or 𝜑 takes place, then the proposition 𝜒 holds. Proposition 47 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜓𝜒) → ((𝜑𝜒) → 𝜒)))

Theoremfrege48 37972 Closed form of syllogism with internal disjunction. If 𝜑 is a sufficient condition for the occurence of 𝜒 or 𝜓 and if 𝜒, as well as 𝜓, is a sufficient condition for 𝜃, then 𝜑 is a sufficient condition for 𝜃. See application in frege101 38084. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → ((𝜒𝜃) → ((𝜓𝜃) → (𝜑𝜃))))

Theoremfrege49 37973 Closed form of deduction with disjunction. Proposition 49 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜑𝜒) → ((𝜓𝜒) → 𝜒)))

Theoremfrege50 37974 Closed form of jaoi 394. Proposition 50 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜓) → ((¬ 𝜑𝜒) → 𝜓)))

Theoremfrege51 37975 Compare with jaod 395. Proposition 51 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜃𝜒) → (𝜑 → ((¬ 𝜓𝜃) → 𝜒))))

20.27.3.5  _Begriffsschrift_ Chapter II with logical equivalence

Here we leverage df-ifp 1013 to partition a wff into two that are disjoint with the selector wff.

Thus if we are given (𝜑 ↔ if-(𝜓, 𝜒, 𝜃)) then we replace the concept (illegal in our notation ) (𝜑𝜓) with if-(𝜓, 𝜒, 𝜃) to reason about the values of the "function." Likewise, we replace the similarly illegal concept 𝜓𝜑 with (𝜒𝜃).

Theoremaxfrege52a 37976 Justification for ax-frege52a 37977. (Contributed by RP, 17-Apr-2020.)
((𝜑𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))

Axiomax-frege52a 37977 The case when the content of 𝜑 is identical with the content of 𝜓 and in which a proposition controlled by an element for which we substitute the content of 𝜑 is affirmed ( in this specific case the identity logical funtion ) and the same proposition, this time where we substituted the content of 𝜓, is denied does not take place. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))

Theoremfrege52aid 37978 The case when the content of 𝜑 is identical with the content of 𝜓 and in which 𝜑 is affirmed and 𝜓 is denied does not take place. Identical to biimp 205. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜑𝜓))

Theoremfrege53aid 37979 Specialization of frege53a 37980. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → ((𝜑𝜓) → 𝜓))

Theoremfrege53a 37980 Lemma for frege55a 37988. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(if-(𝜑, 𝜃, 𝜒) → ((𝜑𝜓) → if-(𝜓, 𝜃, 𝜒)))

Theoremaxfrege54a 37981 Justification for ax-frege54a 37982. Identical to biid 251. (Contributed by RP, 24-Dec-2019.)
(𝜑𝜑)

Axiomax-frege54a 37982 Reflexive equality of wffs. The content of 𝜑 is identical with the content of 𝜑. Part of Axiom 54 of [Frege1879] p. 50. Identical to biid 251. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝜑𝜑)

Theoremfrege54cor0a 37983 Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜓𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))

Theoremfrege54cor1a 37984 Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if-(𝜑, 𝜑, ¬ 𝜑)

Theoremfrege55aid 37985 Lemma for frege57aid 37992. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (𝜓𝜑))

Theoremfrege55lem1a 37986 Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜏 → if-(𝜓, 𝜑, ¬ 𝜑)) → (𝜏 → (𝜓𝜑)))

Theoremfrege55lem2a 37987 Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))

Theoremfrege55a 37988 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))

Theoremfrege55cor1a 37989 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜓𝜑))

Theoremfrege56aid 37990 Lemma for frege57aid 37992. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (𝜑𝜓)) → ((𝜓𝜑) → (𝜑𝜓)))

Theoremfrege56a 37991 Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))))

Theoremfrege57aid 37992 This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri 218. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜓𝜑))

Theoremfrege57a 37993 Analogue of frege57aid 37992. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))

Theoremaxfrege58a 37994 Identical to anifp 1020. Justification for ax-frege58a 37995. (Contributed by RP, 28-Mar-2020.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))

Axiomax-frege58a 37995 If 𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2352. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))

Theoremfrege58acor 37996 Lemma for frege59a 37997. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜓𝜒) ∧ (𝜃𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))

Theoremfrege59a 37997 A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of [Frege1879] p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 37933 incorrectly referenced where frege30 37952 is in the original. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

(if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓𝜒) ∧ (𝜃𝜏))))

Theoremfrege60a 37998 Swap antecedents of ax-frege58a 37995. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))

Theoremfrege61a 37999 Lemma for frege65a 38003. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
((if-(𝜑, 𝜓, 𝜒) → 𝜃) → ((𝜓𝜒) → 𝜃))

Theoremfrege62a 38000 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2562 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(if-(𝜑, 𝜓, 𝜃) → (((𝜓𝜒) ∧ (𝜃𝜏)) → if-(𝜑, 𝜒, 𝜏)))

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